To our initial surprise, the results for Vertex Cover cannot be generalized to Feedback Vertex Set.
It turns out that a minor-closed graph family \(\mathcal {G}\)  must have bounded elimination distance to a forest (Definition REF ), in order to allow a polynomial kernel in a \(\mathcal {G}\) -modulator.
This concept was introduced by Bulian and Dawar [1]} and is another generalization of the more common parameter treedepth [2]}.
Our result is described in Theorem REF .
